Help: analytical of 2nd order PDE

[pangyatou]

What method can I use to analytically solve the following 2nd order PDE?
u=u(x,t)
∂u/∂t - a*x*∂u/∂x-D*∂^{2}u/∂t^{2} = 0
I.C.: u(x,t=0)=u_i
B.C.: u(x=+∞)=0
u(x=-∞)=1

Is self-similar the only way to solve it, or is there any other method can be used to solve it?
How to set the similarity variable if I use self-similar method?

Thanks

 

[pangyatou]

I use separation variable:
u(x,t)=F(η)T(t): η=sqrt(a/D) x
Then obtain the following 2 ODEs:
(1)T'+a λ T=0
(2)F"+η F'+λ F=0

How to solve the second ODE?